General Martingale Research Articles

A general martingale approach to large noise homogenization

A general martingale approach to large noise homogenization

A Berry–Esseen bound of order $\protect \frac{1}{\protect \sqrt{n}} $ for martingales

Renz [13] has established a rate of convergence 1/n in the central limit theorem for martingales with some restrictive conditions. In the present paper a modification of the methods, developed by Bolthausen [2] and Grama and Haeusler [6], is applied for obtaining the same convergence rate for a class of more general martingales. An application to linear processes is discussed.

Burkholder\u2013Davis\u2013Gundy Inequalities in UMD Banach Spaces

In this paper we prove Burkholder–Davis–Gundy inequalities for a general martingale M with values in a UMD Banach space X. Assuming that M_0=0, we show that the following two-sided inequality holds for all 1le p<infty : Here gamma ([![M]!]_t) is the L^2-norm of the unique Gaussian measure on X having [![M]!]_t(x^*,y^*):= [langle M,x^*rangle , langle M,y^*rangle ]_t as its covariance bilinear form. This extends to general UMD spaces a recent result by Veraar and the author, where a pointwise version of (star ) was proved for UMD Banach functions spaces X. We show that for continuous martingales, (star ) holds for all 0<p<infty , and that for purely discontinuous martingales the right-hand side of (star ) can be expressed more explicitly in terms of the jumps of M. For martingales with independent increments, (star ) is shown to hold more generally in reflexive Banach spaces X with finite cotype. In the converse direction, we show that the validity of (star ) for arbitrary martingales implies the UMD property for X. As an application we prove various Itô isomorphisms for vector-valued stochastic integrals with respect to general martingales, which extends earlier results by van Neerven, Veraar, and Weis for vector-valued stochastic integrals with respect to a Brownian motion. We also provide Itô isomorphisms for vector-valued stochastic integrals with respect to compensated Poisson and general random measures.

Fractional integrals and their commutators on martingale Orlicz spaces

On the dyadic martingales, Chao and Ombe (1985) proved the Lp-Lq boundedness of the fractional integrals Iα and their commutators [b,Iα], where b is in BMO or Lipschitz spaces. We extend these results to LΦ-LΨ boundedness of generalized fractional integrals Iγ and their commutators [b,Iγ], where LΦ and LΨ are Orlicz spaces and b is in generalized martingale Campanato spaces on more general martingales. We also prove the LΦ-FLΨϕ boundedness of [b,Iγ], where FLΨϕ is the Triebel-Lizorkin-Orlicz space. Moreover, we characterize b such that the commutator is bounded or compact.

Martingale optimal transport in the discrete case via simple linear programming techniques

We consider the problem of finding consistent upper price bounds and super replication strategies for exotic options, given the observation of call prices in the market. This field of research is called model-independent finance and has been introduced by Hobson 1998. Here we use the link to mass transport problems. In contrast to existing literature we assume that the marginal distributions at the two time points we consider are discrete probability distributions. This has the advantage that the optimization problems reduce to linear programs and can be solved rather easily when assuming a general martingale Spence Mirrlees condition. We will prove the optimality of left-monotone transport plans under this assumption and provide an algorithm for its construction. Our proofs are simple and do not require much knowledge of probability theory. At the end we present an example to illustrate our approach.

Variational estimates for martingale paraproducts

We show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053).

Existence and uniqueness results for BSDE with jumps: the whole nine yards

This paper is devoted to obtaining a wellposedness result for multidimensional BSDEs with possibly unbounded random time horizon and driven by a general martingale in a filtration only assumed to satisfy the usual hypotheses, i.e. the filtration may be stochastically discontinuous. We show that for stochastic Lipschitz generators and unbounded, possibly infinite, time horizon, these equations admit a unique solution in appropriately weighted spaces. Our result allows in particular to obtain a wellposedness result for BSDEs driven by discrete--time approximations of general martingales.

Jump-diffusion processes in random environments

In this paper we investigate jump-diffusion processes in random environments which are given as the weak solutions of SDEs. We formulate conditions ensuring existence and uniqueness in law of solutions. We investigate the Markov property. To prove uniqueness we solve a general martingale problem for càdlàg processes. This result is of independent interest. Application of our results to generalized exponential Lévy model are present in the last section.

Martingale inequalities and deterministic counterparts

We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operator involving a concave envelope. Our results yield an explanation for certain inequalities that arise in mathematical finance in the context of robust hedging.

General Martingale Characterization of G-Brownian Motion

The objective of this article is to derive a general martingale characterization of G-Brownian motion, which generalizes the results obtained in Xu [17]. For this end, we first study some extensions of stochastic calculus with respect to G-martingales under the sublinear expectation spaces.

On equivalency of martingales and related problems

We study some almost everywhere convergence problems for martingales. We establish various equivalency theorems, which show that in some problems of martingales theory the general martingales can be replaced by Haar martingales. Some applications of the obtained results to the theory of differentiation of integrals and convergence of Riemann sums are also discussed.

The Kurzweil-Henstock theory of stochastic integration

The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Ito integral, and a simpler and more direct proof of the Ito Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock’s Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price process S follows a general martingale. This is equivalent to studying the first centered absolute moment of S. We show that if S has a continuous part, the leading term is of order √T in time T and depends only on the initial value of the volatility. Furthermore, the term is linear in T if and only if S is of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation of S so that calculations are necessary only for the class of Lévy processes.

Small-Time Asymptotics of Option Prices and First Absolute Moments

We study the leading term in the small-time asymptotics of at-the-money call option prices when the stock price processSfollows a general martingale. This is equivalent to studying the first centered absolute moment ofS. We show that ifShas a continuous part, the leading term is of order √Tin timeTand depends only on the initial value of the volatility. Furthermore, the term is linear inTif and only ifSis of finite variation. The leading terms for pure-jump processes with infinite variation are between these two cases; we obtain their exact form for stable-like small jumps. To derive these results, we use a natural approximation ofSso that calculations are necessary only for the class of Lévy processes.

Statistical model uncertainty for state-observation models driven by diffusion process

This paper is motivated by the fact that model uncertainty is common-seen in statistics and applied probability. As a response, a likelihood ratio-based approach is presented to statistical detection within a class of state-observation models. Here, the system is driven by some diffusion process while the continuous-time observation process is of additive white noise. The proposed approach can be implemented recursively to compare the competing stochastic systems by fitting the observed historical data. It is superior to the traditional hypothesis test in both theoretical and numerical senses. Moreover, it can be extended to the more general martingale problem setup.

General theories unifying ergodic averages and martingales

The curious fact that the behavior of ergodic averages is analogous to the behavior of (reversed) martingales has long been known. For the last 60 years, at least five different approaches have been proposed to the construction of a unifying theory: by M. Jerison (1955), G.-C. Rota (1961), A. and C. Ionescu Tulcea (1963), A.M. Vershik (1960s), and A.G. Kachurovskii (1998). The aim of the present survey is to carry out a comparative analysis of all the approaches, with special focus being placed on the fifth approach, which is due to the present author and has not yet been published in full detail. Moreover, the comparative analysis carried out in the survey has led to a new, the sixth, approach.

Martingale integrals over Poissonian processes and the Ito-type equations with white shot noise.

The construction of the Ito-type stochastic integrals and differential equations for compound Poisson processes is provided. The general martingale and nonanticipating properties of the ordinary (Gaussian) Ito theory are conserved. These properties appear particularly important if the stochastic description has to be proposed according to game theory or the linear relaxation (or the exponential growth) requirements. In contrast to the ordinary Ito theory the (uncorrelated) parametric fluctuation of a definite sign can be still modeled by asymmetric white shot noise, so the general scope of applications is not restricted by the positivity requirements. The possible use of the developed formalism in econophysics is addressed.

Embedding and asymptotic expansions for martingales

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these results is the establishment of second (and in some cases higher) order asymptotic expansions for martingales.

Time-space polynomial martingales cenerated by a discrete-time martingale

We investigate, for a given martingaleM={M n: n≥0}, the conditions for the existence of polynomialsP(·,·) of two variables, “time” and “space,” and of arbitrary degree in the latter, such that{P(n, M n)} is a martingale for the natural filtration ofM. Denoting by ℘ the vector space of all such polynomials, we ask, in particular, when such a sequence can be chosen so as to span ℘. A complete necessary and sufficient condition is obtained in the case whenM has independent increments. For generalM, we obtain a necessary condition which entails, under mild additional hypotheses, thatM is necessarily Markovian. Considering a slightly more general class of polynomials than ℘ we obtain necessary and sufficient conditions in the case of general martingales also. It is moreover observed that in most of the cases, the set ℘ determines the law of the martingale in a certain sense.

Energy bounds for nonlinear dissipative stochastic differential equations with respect to semimartingales

Stochastic differential equations are investigated under various hypotheses on dissipative and external driving forces. It is proved that global solutions exist and estimates are obtained for bounds on the growth of the energy. New methods are introduced to incorporate general Stieltjes measures and martingales, as well as to treat the conditions of polynomial-exponential bounds on the coefficients of the stochastic differential equations